The present invention is generally related to separating individual source signals from a mixture of source signals, and more specifically related to blind source separation.
A classic problem in signal processing, often referred to as blind source separation (BSS), involves recovering individual source signals from a composite signal comprising a mixture of those individual signals. An example is the familiar xe2x80x9ccocktail partyxe2x80x9d effect, wherein a person at a party is able to separate a single voice from the combination of all voices in the room. The separation is referred to as xe2x80x9cblindxe2x80x9d because it is often performed with limited information about the signals and the sources of the signals.
Blind source separation (BSS) is particularly applicable to cellular and personal wireless communications technologies, wherein many frequency bands have become cluttered with numerous electromagnetic emitters, often co-existing in the same spectrum. The problem of co-channel emitters is expected to only worsen in years to come with the development of low power, unlicensed wireless technologies such as Bluetooth(copyright) and other personal area networks. These developments have resulted in the use of multiple sensors and array signal processing techniques to perform spectral monitoring. Such techniques enable the exploitation of spatial information to separate co-channel emitters for detection, classification, and identification. Additionally, many signals designed for a low probability of detection (LPD) or low probability of intercept (LPI) may use ambient background electromagnetic radiation and known co-channel emitters as a means of concealment. Constructing single sensor receiver systems with the required sensitivity to such emitters is generally prohibitive. Thus, many applications utilize BSS and sensor arrays.
Several techniques have been proposed to solve the BSS problem. These can be classified into two main groups. The first being based on second-order statistics, and the second being based on higher-order statistics, such as those based on independent components analysis (ICA) and other higher-order spectral estimation and spatial filtering techniques.
One second-order blind source separation technique is a spectral estimation method that exploits the rotational invariance of the signal subspace to estimate the direction of arrival. This technique known as Estimation of Signal Parameters via Rotational Invariance (ESPRIT) employs pairs of calibrated elements and uses a matrix-pencil formed by the spatial correlation and cross-correlation matrices. See, for example, R. Roy, A. Paulraj, T. Kailath, xe2x80x9cDirection-of-Arrival Estimation by Subspace Rotation Methods,xe2x80x9d Proc. ICASSP86, pp. 2495-2498 and R. Roy and T. Kailath, xe2x80x9cESPRITxe2x80x94Estimation of Signal Parameters via Rotational Invariance Techniques,xe2x80x9d IEEE Trans. on ASSP, Vol. 37, No. 7, July 1989, pp. 984-995, which are each incorporated by reference in their entirety as if presented herein. However, a disadvantage of ESPRIT is that at very low signal-to-noise ratios the signal plus noise subspace and the noise subspace are indistinguishable thus making the conversion of the noise subspace of the spatial correlation matrix impractical. This is due in part to ESPRIT requiring an estimation of the noise variance to convert the noise subspace into a null subspace of the spatial correlation matrix, and assuming that the noise is spatially white.
Another second order blind source separation technique is known as the Constant Modulus Algorithm (CMA), also referred to as Goddard""s algorithm. The CMA is an adaptive spatial filtering technique, which is used to perform source separation by determining a set of spatial filter tap weights that forces the output signal to have a modulus as close to unity as possible. Typically, the CMA is performed sequentially to separate all source signals. The CMA has been suggested for use as a blind equalization technique to reduce inter-symbol interference of constant modulus signals, such as FSK, PSK, and FM modulated signals, on telephone channels (See, for example, D. N. Godard, xe2x80x9cSelf-recovering Equalization and Carrier Tracking in Two-dimensional Data Communication Systems,xe2x80x9d IEEE Trans. Commun., Vol. COMM-28, November 1980, pp. 1867-1875, which is incorporated by reference in its entirety as if presented herein.), and to perform blind equalization to combat multi-path fading and to suppress co-channel interfering signals (See, for example, B. G. Agee, xe2x80x9cThe Property Restoral Approach to Blind Adaptive Signal Extraction,xe2x80x9d Ph.D. Dissertation, Dept. Elect. Eng. Comput. Science, Univ. of Calif., Davis, 1989, which is incorporated by reference in its entirety as if presented herein). However, the CMA technique works only for signals with a constant modulus and is not practicable for most applications. In practice, because signals are filtered to limit their spectral occupancy at the transmitter and to limit the noise bandwidth at the receiver, true constant modulus signals rarely exist. Furthermore, at very low signal-to-noise ratios, noise dominates the input signal thus distorting the spatial filter""s output signal""s modulus and causing large fluctuation in the error signal used in adaptation.
Yet another second order blind source separation technique is a spatial filtering technique using second-order cyclostationary statistics with the assumption that the source signals are cyclostationary. This technique was developed as a blind single-input single-output (SISO) channel identification technique for use in blind equalization (See, for example, L. Tong, G. Xu, and T. Kailath, xe2x80x9cBlind Identification and Equalization Based on Second-Order Statistics: A Time-Domain Approach,xe2x80x9d IEEE Trans. Information Theory, Vol. 40, No. 2, March 1994, pp. 340-349, which is incorporated by reference in its entirety as if presented herein), and was later adapted to perform the blind separation of cyclostationary signals (See, for example, L. Castedo and A. R. Figueiras-Vidal, xe2x80x9cAn Adaptive Beamforming Technique Based on Cyclostationary Signal Properties,xe2x80x9d IEEE Trans. Signal Processing, Vol. 43, No. 7, July 1995, pp. 1637-1650, which is incorporated by reference in its entirety as if presented herein). One disadvantage of this cyclostationary approach is that it requires different symbol rates and/or different carrier frequencies for separating multiple superimposed signals. Another disadvantage is that residual carrier offsets with a random initial phase can cause the signals to become stationary, causing the cyclostationary assumption to become invalid. Other disadvantages include the fact that this approach precludes separating sources that may use non-linear or non-digital modulations, and this approach assusmes the noise vector is temporally and spatially white.
Still another blind source separation technique based on second-order statistics is referred to as Second-Order Blind Identification (SOBI). See, for example, A. Belouchrani, K. Abed-Meraim, J. F. Cardoso, and E. Moulines, xe2x80x9cBlind Source Separation Using Second-Order Statistics,xe2x80x9d IEEE Trans. Signal Processing, Vol. 45, No. 2, February 1997, pp. 434-444, for a description of this technique, which is incorporated by reference in its entirety as if presented herein. This technique exploits the time coherence of the source signals and relies on the joint diagonalization of a set of covariance matrices. A disadvantage of this technique is that it requires additive noise to be temporally white and uses the eigenvalues of the zero lag matrix to estimate the noise variance and to spatially whiten the sensor output vector. Another disadvantage is that at low signal to noise rations, the estimation of the noise variance is difficult at best and impossible in most cases. Yet another disadvantage is that the number of sources must be known or estimated. Still, another disadvantage is that the SOBI technique is valid for spatially correlated noise. The estimation of the noise covariance is extremely difficult even at high signal-to-noise ratios, thus making the technique impracticable.
Another second-order blind source separation technique is based on the generalized eigen decomposition of a spatial covariance matrix-pencil. This technique is related to the ESPRIT algorithm, but does not require 2N sensors to separate up to N-1 signals because time and/or polarization diversity is used to estimate the pair of spatial covariance matrices. This technique uses a dual polarized array and estimates the spatial covariance matrices on the two orthogonal polarizations to form the matrix pencil. See, for example, A. Belouchrani, K. Abed-Meraim, J. F. Cardoso, and E. Moulines, xe2x80x9cBlind Source Separation Using Second-Order Statistics,xe2x80x9d IEEE Trans. Signal Processing, Vol. 45, No. 2, February 1997, pp. 434-444, which is incorporated by reference in its entirety as if presented herein. This later evolved into using spatial covariance matrices at a zero time lag and a non-zero time lag to form the matrix-pencil. One disadvantage of this technique is that it is limited to separating up to N-1 sources with N sensors. This is due in part to the approach requiring the estimation of the noise variance, similar to ESPRIT, and assuming that the noise is spatially and temporally white.
Finally, another second order blind source separation technique utilizes two non-zero time lags in the estimation of the spatial covariance matrices. See, for example, C. Chang, Z. Ding, S. F. Yau, and F. H. Y. Chan, xe2x80x9cA Matrix-Pencil Approach to Blind Separation of Non-White Sources in White Noise,xe2x80x9d Proc. ICASSP98, Vol. IV, pp. 2485-2488, and C. Chang, Z. Ding, S. F. Yau, and F. H. Y. Chan, xe2x80x9cA Matrix-Pencil Approach to Blind Separation of Colored Non-Stationary Signals,xe2x80x9d IEEE Trans. Signal Processing, Vol. 48, No. 3, March 2000, pp. 900-907, which are each incorporated by reference in their entirety as if presented herein. The non-zero time lags combined with the assumption that the noise vector is temporally white eliminates the need to estimate the noise variance(s) in order to remove the noise subspace and thus allows the technique to separate up to N sources with N sensors. However, disadvantages of the this second-order matrix-pencil technique include the requirement that the noise vector be temporally white and the fact that the estimate of the separation matrix is not done with one of the spatial covariance matrices at a zero time lag, which is when the signal auto-correlations are at their maximum values. These disadvantages are exacerbated by the fact that in many practical applications the noise bandwidth is limited to be on the order of the signal bandwidth making the noise and signal decorrelation times to be approximately equal.
Higher-order blind source separation techniques include all methods that employ statistics of order greater than two. These include independent component analysis (ICA) methods, spatial filtering methods, and spectral estimation based methods and can use either moments or cumulants of order three or higher.
The independent components analysis (ICA) methods seek a separating matrix that maximizes the statistical independence of the outputs of the separation process. See, for example, A. K. Nandi, Blind Estimation Using Higher-Order Statistics. (Kluwer Academic, Dordecht, The Netherlands: 1999), A. Hyvxc3xa4rinen, xe2x80x9cSurvey on Independent Component Analysis,xe2x80x9d Neural Computing Surveys, Vol. 2, No. 1, 1999, pp. 94-128, and J. F. Cardoso, xe2x80x9cBlind Signal Separation: Statistical Principles""xe2x80x9d Proc. of the IEEE, Vol. 9, No. 10, October 1998, pp.2009-2025, which are each incorporated by reference in their entirety as if presented herein. Various ICA methods are known, and are primarily differentiated by the objective/contrast function used to measure statistical independence. Examples of ICA based blind source separation algorithms include the Jxc3xctten-Herault algorithm (See, for example, C. Jxc3xctten and J. Herault, xe2x80x9cBlind Separation of Sources, Part I: An Adaptive Algorithm Based on Neuromimetic Architectures,xe2x80x9d Signal Processing, Vol. 24, 1991, pp. 1-10, which is incorporated by reference in its entirety as if presented herein), which attempts to achieve separation via canceling non-linear correlations through the use of a neural network; the higher-order eigenvalue decomposition or HOEVD method (See, for example, P. Comon, xe2x80x9cIndependent Component Analysis, A New Concept?,xe2x80x9d Signal Processing, Vol. 36, No. 3, April 1994, pp. 287-314, which is incorporated by reference in its entirety as if presented herein); the joint approximate diagonalization of eigen matrices (JADE) algorithm (See, for example, J. F. Cardoso and A. Souloumiac, xe2x80x9cBlind Beamforming for Non-Gaussian Signals,xe2x80x9d IEE Proceedings F, Vol. 140, No. 6, December 1993, pp. 362-370, which is incorporated by reference in its entirety as if presented herein), which exploits the eigen structure of the fourth-order cumulant tensor; the information maximization or infomax technique (See, for example, A. J. Bell and T. J. Sejnowski, xe2x80x9cAn Information-Maximization Approach to Blind Source Separation and Blind Deconvolution,xe2x80x9d Neural Computing, Vol. 7, 1995, pp. 1 129-1159, which is incorporated by reference in its entirety as if presented herein), which seeks a separation matrix that maximizes the output entropy; and the equivariance adaptive source separation or EASI algorithm (See, for example, J. F. Cardoso and B. Hvam Laheld, xe2x80x9cEquivariance Adaptive Source Separation,xe2x80x9d IEEE Trans. On Signal Processing, Vol. 44, No. 12, December 1996, pp. 3017-3030, which is incorporated by reference in its entirety as if presented herein), in which an estimate of the mixing matrix is chosen such that transformations in the sensor output data produces a similar transformation in the estimated mixing matrix. Disadvantages of these ICA techniques include (1) a pre-whitening step is required, (s), the techniques need to be parametrically tuned to the application, and (3) the ICA based techniques tend to have a slow convergence time.
Another higher-order blind source separation technique, known as the Kurtosis Maximization Algorithm (KMA), utilizes kurtosis as a separation measure (See, for example, Z. Ding, xe2x80x9cA New Algorithm for Automatic Beamforming,xe2x80x9d Proc. 25th Asilomar Conf. Signals, Syst., Comput., Vol. 2, 1991, pp. 689-693, and Z. Ding and T. Nguyen, xe2x80x9cStationary Points of Kurtosis Maximization Algorithm for Blind Signal Separation and Antenna Beamforming,xe2x80x9d IEEE Trans. Signal Processing, Vol. 48, No. 6, June 2000, pp. 1587-1596, which is incorporated by reference in its entirety as if presented herein). The KMA is an adaptive spatial filtering technique with a non-deterministic convergence time. One disadvantage of the KMA is that it can not simultaneously separate the source signals. The KMA requires that the sources be separated sequentially one at a time, starting with the signal having the largest kurtosis. Thus, no knowledge of the number of signals is provided by the technique. Other disadvantage of the KMA are that it requires the noise to be spatially white and its convergence with multiple signals has only been proven in the noise free case.
Finally, another higher-order BSS technique is a higher-order version of the ESPRIT algorithm. As the name implies, higher-order ESPRIT replaces the spatial correlation or covariance matrices with spatial fourth-order cumulant matrices. See H. H. Chiang and C. L. Nikias, xe2x80x9cThe ESPRIT Algorithm with Higher-Order Statistics,xe2x80x9d Proc. Workshop on Higher-Order Spectral Analysis, Vail, CO., June 1989, pp. 163-168, C. L. Nikias and A. P. Petropulu, Higher-Order Spectra Analysis: A Non-Linear Signal Processing Framework. (PTR Prentice-Hall, Upper Saddle River, N.J.: 1993), and M. C. Dogan and J. M. Mendel, xe2x80x9cApplications of Cumulants to Array Processingxe2x80x94Part I: Aperture Extension and Array Calibration,xe2x80x9d IEEE Trans. Signal Processing, Vol. 43, No. 5, May 1995, pp. 1200-1216, for descriptions of various types of higher-order ESPRIT techniques, which are each incorporated by reference in their entirety as if presented herein. These higher-order ESPRIT techniques possess several disadvantages. Some of these higher-order ESPRIT techniques require the calibration of the N pairs of sensors but can now separate up to N sources since the noise variance no longer needs to be estimated. Other higher-order ESPRIT techniques require the array to be calibrated to guarantee that the pairs of sensors have identical manifolds (similar to the standard ESPRIT). These techniques degrade in performance significantly as the sensor pairs"" manifolds deviate from one another.
Each of the above-mentioned blind source separation techniques have the disadvantages noted. Additionally, none of the above-mentioned blind source separation techniques operate satisfactorily in the situation where there is a low signal-to-noise plus interference ratio. Accordingly, an improved blind source separation technique is desired.
In one embodiment of the present invention, a method for separating a plurality of signals provided by a respective plurality of sources and received by an array comprising a plurality of elements, includes generating a separation matrix as a function of time differences between receipt of the plurality of signals by the plurality of elements and a spatial fourth order cumulant matrix pencil. The method also includes multiplying the separation matrix by a matrix representation of the plurality of signals.
In another embodiment of the present invention, a system for separating a plurality of signals provided by a respective plurality of sources includes a receiver for receiving the plurality of signals and for providing received signals. The system also includes a signal processor for receiving the received signals, generating a separation matrix, and multiplying the separation matrix by a matrix representation of the received signals. The separation matrix is a function of time differences between receipt of the plurality of signals by the receiver and a function of a spatial fourth order cumulant matrix pencil.